Characterizing population dynamics How fast does a population increase/decrease? (rate of change) What will the population size be next year? (management) Need to summarize Births & Deaths (age structure) of a population Cohort Life Tables follow one group from birth until the last one dies Static Life Table census population for abundance in each age/stage combined with estimates of survival and reproductive output by age/stage Limitations? Must be able to age/stage each organism (challenging, uncertainty)
Survivorship curves
Structured demographic models Translate variation in individuals (measureable) to population dynamics (less measurable) Structured around life-history schedules of different species or populations Demography (vital rates by age, sex, stage, size, etc.) captures both the dynamics & structure of populations Widely adopted to evaluate species vulnerability (Population Viability Analysis, late 1990’s )
Life history components Maturity- age at 1st reproduction Mosquito (14 days), Desert tortoise (25-30 years) Parity- # of episodes for reproduction Sockeye salmon (1), White footed mouse (4-12) Semelparity vs. Iteroparity (annual, perennial) Fecundity- # offspring/episode Elephant (1), Western Toad (8,000-15,000) Aging/Senescence- survival/life span - Fruit fly (35 d), Blue Whale (80-90 y)
Not all combinations are possible... Log 2 eggs x 2 / yr 20d parental care Lifespan 2-3yr Log 1 egg/ ~2yr 9 mo. parental care Lifespan >60yr
Life history of Cascades frogs 4 life stages: Embryonic: lasts 1-3 weeks, 0-80 (~60)% survival Larval (tadpole): lasts 8-12 weeks ~50% survival Juvenile (metamorph): lasts 2-3 years (10-40??)% annual survival Adult: lasts 8-20+ years, 300-700 offspring/yr 70-80% annual survival Rana cascade
Life history diagram for Cascades frogs Vital rates, also called transition probabilities 300-700/yr Embryos Larvae Juveniles Adults x 1 - x 0.6 0.5 0.7-0.8 0.1-0.4 2-3 yr 4-7 yr 1 yr
Life history of elephants 5 life stages: Yearling: 80% survival, lasts 1 year Pre-reproductive: 98% survival, lasts 15 year Early reproductive: 98% survival, lasts 5 years, 0.08 offspring/yr Middle reproductive: 95% survival, lasts 25 years, 0.3 offspring/yr Post-reproductive: 80% survival, 5-25 years
Life history diagram for elephants 0.1/yr 0.08/yr 0.8 0.98 0.95 Post-reproductive Pre-reproductive Middle age Early Repro Yearling x x x 1- x 1- x 1- x 1 yr 15 yr 5 yr 25 yr 5-25 yr
Other ways to depict life cycle Age-based Stage-based Size-based
Life histories Bubble diagrams summarize average life history events with fixed time steps (survival per week, year, decade) Result of natural selection Organisms exist to maximize lifetime reproductive success Represent successful ways of allocating limited resources to carry out various functions of living organisms Survival, growth, reproduction
Questions we can answer with age/stage structured population models: How much harvest of a population can occur while still have less than an X% chance of extinction? What life history stages should conservation (or eradication) efforts be focused on to achieve the biggest change in population size/growth rate? How many populations of a species need be preserved to ensure reasonable protection from periodic local extinctions and infrequent catastrophic events? Which life-history strategies are more or less vulnerable to exploitation and extinction? Is it worth the effort to try and recover a particular population, or is it so likely to go extinct that limited resources should be invested elsewhere? and hundreds more!
Single species population growth models
Matrix population models
Basic matrix construction 1 2 3 a21 a32 a33 a13 aij transition prob. to row i from column j (per timestep) Columns = j (from) Rows = i (to)
Basic matrix construction 1 2 3 a21 a32 a33 a13 If there were only one class, what would this look like? x Matrix (A) x population vector (nt) = population vector (nt+1) Underlying this model is an assumption of EXPONENTIAL growth
3 x 3 age-structured matrix (also called Leslie matrix) P=probability of surviving from one age to the next F=fecundity of individuals at each age In this case, there are two pre-reproductive years (maturity at age three) *only need one subscript here because indivds must move ages each time step See any inconsistency here?
4 x 4 size-structured matrix (also called Lefkovitch matrix) Pij=probability of growing from one size to the next or remaining the same size (need subscripts to denote new possibilities) F=fecundity of individuals at each size In this case, there are three pre-reproductive sizes (maturity at age four). **additional complexities like shrinking or moving more than one class back or forward is easy to incorporate
Matrix multiplication 1 2 3 Semipalmated sandpiper data from Hitchcock & Gatto-Trevor (1997) Three stages (1,2,3+ yrs), with reproduction at each stage. a21 a32 a33 1 2 3 x nt per ha Average matrix 1 2 3
Matrix multiplication 1 2 3 x nt How do we calculate the number present in each class next year (nt+1)? =
Matrix multiplication nt n(1) n(2) n(3) n(4) n(5) n(6) n(7) n(8)
Matrix multiplication nt n(1) n(2) n(3) n(4) n(5) n(6) n(7) n(8) Is this population declining/increasing/stable?
Matrix multiplication nt n(1) n(2) n(3) n(4) n(5) n(6) n(7) n(8) = sum (n2) / sum(n1) What’s the proportional change in the population from one time to the next? = lambda (λ) (0.639)
Matrix multiplication Semipalmated sandpiper data from Hitchcock & Gatto-Trevor (1997) nt Average matrix Is this population declining/increasing/stable? Dominant eigenvalue (1) *stable environment or average matrix* Asymptototic measure of geometric, density-independent population growth rate Is the population at a stable distribution of stages/ages? Dominant eigenvector (w)
What to do with a deterministic matrix? Fixed environment assumption is unrealistic. BUT… can evaluate the relative performance of different management/conservation options can use the framework to conduct ‘thought experiments’ not possible in natural contexts can ask whether the results of a short-term experiment/study affecting survival/reproduction could influence population dynamics
A short detour for an example
low UV-B moderate UV-B high UV-B % survival -UV +UV -UV +UV % survival +UV % survival -UV How does egg mortality change across a natural gradient of UV-B exposure? 28
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?? UV 10,000 eggs 3,000 larvae 6,000 larvae 100 juveniles dispersal UV ?? 10,000 eggs 6,000 larvae 3,000 larvae adults (4-10 years) eggs (1-3 weeks) Talk about variability in recruitment to adult population “Population” in most cases includes many breeding sites across the landscape larvae (8 weeks-3+ years) 100 juveniles 30
What to do with a deterministic matrix? Fixed environment assumption is unrealistic. BUT… can evaluate the relative performance of different management/conservation options can use the framework to conduct ‘thought experiments’ not possible in natural contexts can ask whether the results of a short-term experiment/study affecting survival/reproduction could influence population dynamics *can evaluate the relative sensitivity of to different vital rates
Matrix element sensitivities Semipalmated sandpiper data from Hitchcock & Gatto-Trevor (1997) nt Average matrix Reproductive value Sensitivity of 1 to element aij is Sij Prop in j @ stable age constant
Matrix element sensitivities Semipalmated sandpiper data from Hitchcock & Gatto-Trevor (1997) nt Average matrix Sensitivity analysis of a deterministic matrix (by brute force): Vary vital rates individually (small change vs. biologically realistic range?) Re-calculate deterministic 1 Plot change in each rate versus change in 1
Matrix element sensitivities Semipalmated sandpiper data from Hitchcock & Gatto-Trevor (1997) nt Average matrix Sensitivity analysis of a deterministic matrix: X
Matrix element sensitivities Semipalmated sandpiper data from Hitchcock & Gatto-Trevor (1997) nt Average matrix Relative sensitivities (comparable across vital rates) called Elasticities proportional change of vital rates compared to proportional change in lambda Why does might this matter?